3.78 \(\int \frac{(a+b x)^3 (A+B x)}{x} \, dx\)

Optimal. Leaf size=54 \[ a^3 A \log (x)+3 a^2 A b x+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]

[Out]

3*a^2*A*b*x + (3*a*A*b^2*x^2)/2 + (A*b^3*x^3)/3 + (B*(a + b*x)^4)/(4*b) + a^3*A*
Log[x]

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Rubi [A]  time = 0.0499503, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ a^3 A \log (x)+3 a^2 A b x+\frac{3}{2} a A b^2 x^2+\frac{B (a+b x)^4}{4 b}+\frac{1}{3} A b^3 x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/x,x]

[Out]

3*a^2*A*b*x + (3*a*A*b^2*x^2)/2 + (A*b^3*x^3)/3 + (B*(a + b*x)^4)/(4*b) + a^3*A*
Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{3} \log{\left (x \right )} + 3 A a^{2} b x + 3 A a b^{2} \int x\, dx + \frac{A b^{3} x^{3}}{3} + \frac{B \left (a + b x\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x,x)

[Out]

A*a**3*log(x) + 3*A*a**2*b*x + 3*A*a*b**2*Integral(x, x) + A*b**3*x**3/3 + B*(a
+ b*x)**4/(4*b)

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Mathematica [A]  time = 0.0422138, size = 63, normalized size = 1.17 \[ a^3 A \log (x)+\frac{1}{12} x \left (12 a^3 B+18 a^2 b (2 A+B x)+6 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/x,x]

[Out]

(x*(12*a^3*B + 18*a^2*b*(2*A + B*x) + 6*a*b^2*x*(3*A + 2*B*x) + b^3*x^2*(4*A + 3
*B*x)))/12 + a^3*A*Log[x]

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Maple [A]  time = 0.005, size = 70, normalized size = 1.3 \[{\frac{B{b}^{3}{x}^{4}}{4}}+{\frac{A{b}^{3}{x}^{3}}{3}}+B{x}^{3}a{b}^{2}+{\frac{3\,aA{b}^{2}{x}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,{a}^{2}Abx+{a}^{3}Bx+{a}^{3}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x,x)

[Out]

1/4*B*b^3*x^4+1/3*A*b^3*x^3+B*x^3*a*b^2+3/2*a*A*b^2*x^2+3/2*B*x^2*a^2*b+3*a^2*A*
b*x+a^3*B*x+a^3*A*ln(x)

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Maxima [A]  time = 1.33228, size = 92, normalized size = 1.7 \[ \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x,x, algorithm="maxima")

[Out]

1/4*B*b^3*x^4 + A*a^3*log(x) + 1/3*(3*B*a*b^2 + A*b^3)*x^3 + 3/2*(B*a^2*b + A*a*
b^2)*x^2 + (B*a^3 + 3*A*a^2*b)*x

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Fricas [A]  time = 0.19886, size = 92, normalized size = 1.7 \[ \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x,x, algorithm="fricas")

[Out]

1/4*B*b^3*x^4 + A*a^3*log(x) + 1/3*(3*B*a*b^2 + A*b^3)*x^3 + 3/2*(B*a^2*b + A*a*
b^2)*x^2 + (B*a^3 + 3*A*a^2*b)*x

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Sympy [A]  time = 1.33524, size = 73, normalized size = 1.35 \[ A a^{3} \log{\left (x \right )} + \frac{B b^{3} x^{4}}{4} + x^{3} \left (\frac{A b^{3}}{3} + B a b^{2}\right ) + x^{2} \left (\frac{3 A a b^{2}}{2} + \frac{3 B a^{2} b}{2}\right ) + x \left (3 A a^{2} b + B a^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x,x)

[Out]

A*a**3*log(x) + B*b**3*x**4/4 + x**3*(A*b**3/3 + B*a*b**2) + x**2*(3*A*a*b**2/2
+ 3*B*a**2*b/2) + x*(3*A*a**2*b + B*a**3)

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GIAC/XCAS [A]  time = 0.262322, size = 95, normalized size = 1.76 \[ \frac{1}{4} \, B b^{3} x^{4} + B a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + B a^{3} x + 3 \, A a^{2} b x + A a^{3}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x,x, algorithm="giac")

[Out]

1/4*B*b^3*x^4 + B*a*b^2*x^3 + 1/3*A*b^3*x^3 + 3/2*B*a^2*b*x^2 + 3/2*A*a*b^2*x^2
+ B*a^3*x + 3*A*a^2*b*x + A*a^3*ln(abs(x))